Question: $ A = \left[\begin{array}{rr}0 & -1 \\ 0 & -1 \\ -2 & -1\end{array}\right]$ $ E = \left[\begin{array}{rr}-1 & 2 \\ -2 & 0\end{array}\right]$ Is $ A E$ defined?
Solution: In order for multiplication of two matrices to be defined, the two inner dimensions must be equal. If the two matrices have dimensions $( m \times  n)$ and $( p \times q)$ , then $ n$ (number of columns in the first matrix) must equal $ p$ (number of rows in the second matrix) for their product to be defined. How many columns does the first matrix, $ A$ , have? How many rows does the second matrix, $ E$ , have? Since $ A$ has the same number of columns (2) as $ E$ has rows (2), $ A E$ is defined.